Natural generalized inverse and core of an element in semigroups, rings and Banach and Operator Algebras

Details Natural generalized inverse and core of an element in semigroups, rings and Banach and Operator Algebras Natural generalized inverse and core of an element in semigroups, rings and Banach and Operator Algebras

2012-03-16

arxiv.org/abs/1203.3743v1

Mathematics

Group Theory

Scientific paper

Using the recent notion of inverse along an element in a semigroup, and the natural partial order on idempotents, we study bicommuting generalized inverses and define a new inverse called natural inverse, that generalizes the Drazin inverse in a semigroup, but also the Koliha-Drazin inverse in a ring. In this setting we get a core decomposition similar to the nilpotent, Kato or Mbekhta decompositions. In Banach and Operator algebras, we show that the study of the spectrum is not sufficient, and use ideas from local spectral theory to study this new inverse.

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